The Scot has been linked with Chelsea and Tottenham - should, as expected, Harry Redknapp take over at England - while Manchester United boss Sir Alex Ferguson has praised his compatriot for the work he has done at Goodison Park.
Moyes will mark his 10th anniversary at the club on Wednesday still looking for his first trophy and he freely admits he is a hugely ambitious man.
Kenwright is too, but knows the only way to match the ambition of his manager is to find new owners who can bring in much-needed finance for him to be able to compete more regularly with the top-six in the Barclays Premier League.
Asked whether he feared losing Moyes to a wealthier rival, Kenwright said: "Not at all. Never. Why? What do I think is the biggest club in the world? Everton.
"Why wouldn't people want him? But he is our manager and I do believe he is our manager because he wants to be our manager and he gets what he can and what he wants out of this club.
"I think that is important to him. He is ambitious. I am hugely ambitious but I have got to be realistic and that is why I have to ask myself whether I can be the chairman for much longer.
"It is a huge club and I have to find the investment to take it forward.
"We love nights like the ones against Chelsea, Man City and Tottenham (all recently beaten at Goodison Park) but we live in a game dominated by money.
"I know this club is revered throughout football for the way we try to do it but it is a huge job and it gets more and more difficult.
"The one thing he (Moyes) does know is he gets every single brass farthing available and he does know no-one takes any money out of this club.
"But at the end of the day I have to find a new owner."
Of his decade-long relationship with Moyes, Kenwright said there had been very few disagreements.
"It has not been a love-in. Both of us are keen to say that but it has been a friendship," he said.
"It has been 10 years of absolute friendship. We've had maybe two what you could call rows but they would last five minutes, words rather than rows maybe."
However, Kenwright felt the manager had brought something back to Everton which money could not buy.
"The one thing he has done for this club is he has made us all believe again and 10 years ago we had lost some of our belief," he said.
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